The discovery of the two-solar mass neutron stars J1614-2230 (1.97 0.04 M_SUN) and J0348+0432 (2.010.04 M_SUN) allows the possible existence of deconned quarks in the cores of neutron stars. We compute the equation of state of the matter in the cores of hybrid stars for hadronic matter, treatedin the non linear relativistic mean-field approximation and quark matter, represented by three-flavor local and non-local Nambu--Jona-Lasinio (NJL) models with repulsive vector interactions.The transition of hadronic to quark matter (mixed phase) is constructed by considering either a soft phase transition (Gibbs construction) or a sharp phase transition (Maxwell construction). We find that high-mass neutron stars with masses up to 2.1 M_SUN 2.4 M_SUN may contain a mixed phase intheir cores, if global charge conservation is considered through the Gibbs conditions. However, if the Maxwell condition is considered, the appearance of a pure quark matter core either destabilizes thestar immediately (typically for non-local NJL models) or leads to a very short hybrid star branch in the mass-radius relation (usually for local NJL models). Our study also indicates that neutron stars with masses of around 1.4 M_SUN would not contain deconfined quark matter. Very good agreement with the thermal evolution established for the neutron star in Cassiopeia A (Cas A) is obtained forone of our models, if the protons in the core are strongly paired, the repulsion among the quarks is mildly repulsive, and the mass of Cas A has a canonical value of 1.4 M_SUN.